6 research outputs found
A Linear/Producer/Consumer Model of Classical Linear Logic
This paper defines a new proof- and category-theoretic framework for
classical linear logic that separates reasoning into one linear regime and two
persistent regimes corresponding to ! and ?. The resulting
linear/producer/consumer (LPC) logic puts the three classes of propositions on
the same semantic footing, following Benton's linear/non-linear formulation of
intuitionistic linear logic. Semantically, LPC corresponds to a system of three
categories connected by adjunctions reflecting the linear/producer/consumer
structure. The paper's metatheoretic results include admissibility theorems for
the cut and duality rules, and a translation of the LPC logic into category
theory. The work also presents several concrete instances of the LPC model.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441
Modeling Simply-Typed Lambda Calculi in the Category of Finite Vector Spaces
In this paper we use finite vector spaces (finite dimension, over finite fields) as a non-standard computational model of linear logic. We first define a simple, finite PCF-like lambda-calculus with booleans, and then we discuss two finite models, one based on finite sets and the other on finite vector spaces. The first model is shown to be fully complete with respect to the operational semantics of the language, while the second model is not. We then develop an algebraic extension of the finite lambda calculus and study two operational semantics: a call-by-name and a call-by-value. These operational semantics are matched with their corresponding natural denotational semantics based on finite vector spaces. The relationship between the various semantics is analyzed, and several examples based on Church numerals are presented
Entanglement of Sections: The pushout of entangled and parameterized quantum information
Recently Freedman & Hastings asked for a mathematical theory that would unify
quantum entanglement/tensor-structure with parameterized/bundle-structure via
their amalgamation (a hypothetical pushout) along bare quantum (information)
theory. As a proposed answer to this question, we first make precise a form of
the relevant pushout diagram in monoidal category theory. Then we prove that
the pushout produces what is known as the *external* tensor product on vector
bundles/K-classes, or rather on flat such bundles (flat K-theory), i.e., those
equipped with monodromy encoding topological Berry phases. The bulk of our
result is a further homotopy-theoretic enhancement of the situation to the
"derived category" (infinity-category) of flat infinity-vector bundles
("infinity-local systems") equipped with the "derived functor" of the external
tensor product. Concretely, we present an integral model category of simplicial
functors into simplicial K-chain complexes which conveniently presents the
infinity-category of parameterized HK-module spectra over varying base spaces
and is equipped with homotopically well-behaved external tensor product
structure. In concluding we indicate how this model category serves as
categorical semantics for the linear-multiplicative fragment of Linear Homotopy
Type Theory (LHoTT), which is thus exhibited as a universal quantum programming
language. This is the context in which we recently showed that topological
anyonic braid quantum gates are native objects in LHoTT.Comment: 71 pages, various figure
The Quantum Monadology
The modern theory of functional programming languages uses monads for
encoding computational side-effects and side-contexts, beyond bare-bone program
logic. Even though quantum computing is intrinsically side-effectful (as in
quantum measurement) and context-dependent (as on mixed ancillary states),
little of this monadic paradigm has previously been brought to bear on quantum
programming languages.
Here we systematically analyze the (co)monads on categories of parameterized
module spectra which are induced by Grothendieck's "motivic yoga of operations"
-- for the present purpose specialized to HC-modules and further to set-indexed
complex vector spaces. Interpreting an indexed vector space as a collection of
alternative possible quantum state spaces parameterized by quantum measurement
results, as familiar from Proto-Quipper-semantics, we find that these
(co)monads provide a comprehensive natural language for functional quantum
programming with classical control and with "dynamic lifting" of quantum
measurement results back into classical contexts.
We close by indicating a domain-specific quantum programming language (QS)
expressing these monadic quantum effects in transparent do-notation, embeddable
into the recently constructed Linear Homotopy Type Theory (LHoTT) which
interprets into parameterized module spectra. Once embedded into LHoTT, this
should make for formally verifiable universal quantum programming with linear
quantum types, classical control, dynamic lifting, and notably also with
topological effects.Comment: 120 pages, various figure